Let Y be an exponential random variable with parameter $\tau > 0$. Compute the cdf and pdf of $F_W$ where $W = Y^3$

The solution states the cdf as $1 - e^{\frac{-y^\frac{1}{3}}{t}}$ because $F_Y =1 - e^{\frac{y}{\tau}}$. $P[W \leq w] = P[Y^3 \leq w] = P[Y \leq y^{\frac{1}{3}}] = 1 - e^{\frac{-y^\frac{1}{3}}{t}}$

I tried achieving the same results with the following logic: $P[W \leq w] = P[Y^3 \leq w] = P[Y \leq w^{\frac{1}{3}}] = \int_{0}^{w^\frac{1}{3}} \frac{1}{\tau} e^{-\frac{y}{\tau}}dy =\left(e^{-\frac{\sqrt[3]{w}}{\tau}}-1\right)\tau+\frac{\sqrt[3]{w}}{\tau}$

However this does not return the same result. Does anyone know why my method did not return the correct result?

  • $\begingroup$ I don't understand what you're doing during your integration. $\int_0^a be^{-bw}dw=1-e^{-ba}$. Where are you getting $x$ and where are you getting the second term? $\endgroup$ – Alex R. Oct 10 '17 at 22:37
  • $\begingroup$ @AlexR. I've edited my logic a bit. Let me know if there is more confusion $\endgroup$ – Curious Student Oct 10 '17 at 22:42
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    $\begingroup$ Again, you are evaluating the integral incorrectly. $\endgroup$ – Alex R. Oct 10 '17 at 22:52
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    $\begingroup$ That means you entered it incorrectly into symbolab. According to you if $w$ goes to infinity, then probability goes to infinity as well. Does that sound right to you? $\endgroup$ – Alex R. Oct 10 '17 at 22:57
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    $\begingroup$ You should not rely on automatic computation you can't apply reasonableness checks to; otherwise your usage errors turn into gospel. ( (along with the much less likely but still possible presence of bugs). $\endgroup$ – Glen_b -Reinstate Monica Oct 11 '17 at 0:03

First of all,

$$ P[W \le w] = P[Y^3 \le w] = P[Y \le w^{1/3}] = 1- e^\frac{-w^{1/3} }{t}$$

ie. this is a function of W not Y.

Second, as said in the comments your logic is right but your integral is wrong.

We have the expression

$$ \int_0^{w^{1/3}} \frac{1}{t} e^{-y/t} dy $$

Let $$u = \frac{-y}{t} $$

and you get the expression

$$ - \int_0^{-\frac{w^{1/3}}{t}} e^u du$$

Finish this integration and you arrive at the answer.


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